#483414

order	512
nilpotency class	4
isoclinism	isoclinic class of #483412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#427
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂	C₂×#1568
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ #405 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,6,13,8,14,31,43,9,15,30,12,7,34,5,29)(3,16,19,36,21,45,18,37,22,41,32,58,20,54,52,10)(4,23,26,57,28,39,25,42,24,56,33,59,27,11,38,40)(17,46,49,35,51,61,62,55,47,64,53,60,50,44,48,63);(1,3,5,18,7,20,30,19,9,22,31,52,8,21,6,32)(2,10,12,16,14,36,29,45,15,37,13,41,34,58,43,54)(4,17,25,48,27,50,26,53,24,47,38,62,28,51,33,49)(11,35,40,44,23,55,57,46,39,60,42,61,56,63,59,64);(1,4,7,27,9,24,8,28)(2,11,14,23,15,39,34,56)(3,17,20,50,22,47,21,51)(5,25,30,26,31,38,6,33)(10,35,36,55,37,60,58,63)(12,40,29,57,13,42,43,59)(16,44,45,46,41,61,54,64)(18,48,19,53,52,62,32,49);(4,24)(11,39)(17,47)(23,56)(25,38)(26,33)(27,28)(35,60)(40,42)(44,61)(46,64)(48,62)(49,53)(50,51)(55,63)(57,59)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,f^b=fi,d^c=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72217272, #72217274, #72224955, …
maximal normal subgroup of	#72217272, #72217274, #72224955, …
maximal quotient of	C₂×#483414

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390610);